Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [real-numbers]

For questions about $\mathbb{R}$, the field of real numbers. Often used in conjunction with the real-analysis tag.

The field of real numbers, usually denoted by $\mathbb{R}$ or $\mathbf{R}$ is a field equipped with an order, which is complete with respect to that order. Moreover, it is the only ordered field which is complete (up to isomorphism). The real numbers are used as basis for measuring "length".

The real numbers can be classified in various ways: rational and irrational numbers; algebraic and transcendental numbers; computable and non-computable numbers; etc.

The real numbers carry a natural topology, which is generated by the order. The topology can be induced by a naturally arising complete metric. See more on Wikipedia.

4498 questions
2
votes
1 answer

Euclid's Division Lemma Puzzle

This is a folk puzzle I found in one of my Class X textbooks under the Chapter - 'Real Numbers'. This has to be solved somehow using the concepts of Euclid's Division Lemma and LCM. The puzzle goes as follows: A trader was moving along a road…
Aditya Pratap Singh
  • 326
2
votes
2 answers

If only the first compatibility condition of an ordered field is satisfied, then can R have other orders?

The two compatibility conditions for an ordered field F are that: $ \begin{align} x, y, z\in F, & y < z \implies x + y < x + z\\ x, y\in F, & 0 < x, y \implies 0 < xy \end{align} $ Suppose we create an order on the field of real numbers that is…
extremeaxe5
  • 1,110
2
votes
1 answer

Significant figures in 3000.00

This is a dumb question but it came to my mind and I got shocked . If $67.010$ have 4 significant figures because I count the 3rd decimal place into it , then how many significant figures has $3000.00$ has ? Am I right to say 6 ?
user307640
  • 2,632
  • 2
  • 17
  • 27
2
votes
3 answers

Real number * Integer = Integer

Lets say $r$ is a real number and $I_1, I_2$ are integers. I want this, $$r*I_1=I_2$$ This is not possible all the time so, I allow to add an epsilon $\epsilon$ to the real number such that $$(r+\epsilon)*I_1 = I_2$$ What is the smallest value of…
top secret
  • 38
2
votes
2 answers

Real numbers mapped to Sequences of decimal digits

Apologies for the simple question but my google-fu has left me and my mind is weak... Can all arbitrary sequences of decimal digits be put into one to one correspondence with the real numbers? How? Decimal expansion won't work because 1 =…
Christopher Edwards
  • 123
2
votes
4 answers

Countable subsets of the reals

The naturals, integers, rationals, algebraic, computable numbers are all countable subsets of the reals. What are some more interesting esoteric subsets of the reals which are countable? Which of those countable subsets actually form a field?
Joshhh
  • 2,438
2
votes
2 answers

Uniqueness of set of Positive real numbers using order axioms

While constructing the real numbers, we come across the order axioms, used to define the positive real numbers, i.e there exists a set $P$ such that the properties of trichotomy, closure under addition and multiplication hold. The axioms guarantee…
Student
  • 393
2
votes
5 answers

Why does an argument similiar to $\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+...=1$ show that $2+4+8+...=-2$

See how to prove $\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+...=1$ $x=\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+...$ $2x=1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+...$ Then: $x=1$ Now I use the same argument to prove…
Taha Akbari
  • 3,559
2
votes
2 answers

How can I check easily if all numbers in a number set equals to each other or not?

I want to know if all number pair equals each other that selected from a specified number set. For example: There is a set $A=\{5,3,6,2\}$ and to check the variable equalities requires to check the boolean result comparing each number to others.…
Ali Tor
  • 180
2
votes
1 answer

Is there a "jagged" real-valued function that is "smooth" in cardinalities greater than the reals?

My background: I have a bachelor's CS degree and have never taken anything beyond part of a first course in abstract algebra - no real analysis or complex analysis. I learned about higher cardinalities than $R$ in my automata course. The inverse of…
Jesus is Lord
  • 520
2
votes
2 answers

Can any Real number be typed in a computer?

Suppose that we have a computer program, my question is whether a human can type in any real number - say in $[0,10]$ - that she would like to type in a finite amount of time? Suppose that the program allows typing in any basic expressions like sum,…
Nonlinear
  • 31
2
votes
2 answers

Non computable numbers are normals?

We know that almost all real number are normal and almost all real number are non computable. This does not suffice to deduce that all non computable numbers are normals but , intuitively (??) this seems reasonable. There is some proof ( or…
Emilio Novati
  • 62,675
2
votes
2 answers

Find the last digit of the sum $1!+2!+3!+...+49!$

Is there any formula for finding the last digit of the factorials? How to approach these type of questions? Thanks in advance.
PARTH
  • 33
2
votes
4 answers

Proof that $ \forall x,y \in \mathbb{R} \qquad x^2+y^2+(x-1)(y-1)>0 $

How to proof simply that $$ \forall x,y \in \mathbb{R} \qquad x^2+y^2+(x-1)(y-1)>0 $$
Hamza
  • 3,781
1
vote
1 answer

Proof from equality

suppose this equality holds: x, y, z are real 15(x + y + z) = 12(xy + yz + xz) = 10(x^2 + y^2 + z^2 ) and at least one variables isn't zero. I need to proof that x + y + z = 4 and find the smallest posible closed interval [a,b] in which are all…
balast
  • 13
1 2 3
…
20 21